Moving Square to Sit onto Sphere's Face?
3 techniques at a minimum are possible.  

Particles and Duplication on faces and Inset Faces.



Consider particles in the selected object in the image below.

The orange cubes are particle objects rendered by the particle system.



Consider duplicated objects in the select object in the image below.
The selected sphere is the parent of the cyan cube.  Duplication panel ... select faces.



The cubes location are determined by the sphere.  They are moved off the sphere for discussion purposes.

3rd Technique for Trapezoids placement.

Create Cyan UV Sphere. Edit Mode. Mesh Select Mode Faces. Select All.

Inset Faces. Accept Inset. Keep Selection.

Scale Mode to individual Origins.
Scale To Taste.
Switch Mesh Select Mode Vertices. Invert Selection.  Delete Vertices.
Some of this is visible in the Info Window in the image below.

For discussion purposes a yellow sphere was placed in the middle.